Compressed sensing is a rapidly emerging field which proposes a new approach to sample data far below the Nyquist rate when the sampled data admits a sparse approximation in some appropriate dictionary. The approach is supported by many theoretical results on the identification of sparse
representations in overcomplete dictionaries, but many challenges remain open to determine its range of effective applicability.

METISS has chosen to focus more specifically on the exploration of Compressed Sensing of Acoustic Wavefields. This research has began in the framework of the Ph.D. of Prasad Sudhakar (started in december 2007), and we have set up the ANR collaborative project ECHANGE (ECHantillonnage
Acoustique Nouvelle GEnération) which is due to begin in January 2009. Rémi Gribonval is the coordinator of the project.

The main challenges are: a) to identify dictionaries of basic wavefield atoms making it possible to sparsely represent the wavefield in several acoustic scenarios of interest; b) to determine which types of (networks) of acoustic sensors maximise the identifiability of the sparse
wavefield representation, depending on the acoustic scenario; c) to design scalable algorithms able to reconstruct the measured wavefields in a region of interest.

Compressed sensing of wideband signals

Compressed sensing is also the object of a collaboration with EPFL in the framework of the Equipe Associée SPARS
8.1.1 . In the framework of the summer internship of Mr Farid Naini Mohavedian, we studied the application of compressed sensing to ultra wide-band signals. More precisely, we studied a
model where the considered signals are sparse linear superpositions of shifts of a known, potentially wide-band, pulse. This signal model is key for applications such as Ultra Wide Band (UWB) communications or neural signal processing. We compared several acquisition strategies and showed
that the approximations recovered via
1minimization are greatly enhanced if one uses Spread Spectrum analog modulation prior to applying random Fourier measurements. We complemented our experiments with a discussion of possible hardware implementation of our technique, and checked that a simplified hardware implementation
did not degrade the performance of the compressed sensing system. The results have been submitted at the conference ICASSP 2009
[34] .

Wavelets on graphs

Within the framework of the Equipe Associée SPARS
8.1.1 , we investigated the possibility of developping sparse representations of functions defined on graphs, by defining an extension to the traditional wavelet transform which is
valid for data defined on a graph. The transform is based on spectral graph theory and allows the construction of families of multi-scale atoms which are well adapted to the specific connectivity of a graph.

We proved that it is possible to build a wavelet family which constitutes a frame, which is an important property to represent functions (i.e. signals). We also studied certain families of wavelets on graphs which generalize to the multiscale setting the notion of arbitrary powers of the
Laplacian on the graph.

These wavelets could turn out to be very useful in application scenarios where signals are sampled non-uniformly (such as in sensor networks) or when graphs are inherently present in the model (e.g. in social networks). These results will be presented at the workshop "sparsity and large
inverse problems" to be held in Cambridge in December 2008.